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Theorem rfovfvd 40226
Description: Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, and relation 𝑅. (Contributed by RP, 25-Apr-2021.)
Hypotheses
Ref Expression
rfovd.rf 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))
rfovd.a (𝜑𝐴𝑉)
rfovd.b (𝜑𝐵𝑊)
rfovfvd.r (𝜑𝑅 ∈ 𝒫 (𝐴 × 𝐵))
rfovfvd.f 𝐹 = (𝐴𝑂𝐵)
Assertion
Ref Expression
rfovfvd (𝜑 → (𝐹𝑅) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑟,𝑥   𝐵,𝑎,𝑏,𝑟,𝑥   𝑦,𝐵,𝑎,𝑏,𝑟   𝑅,𝑟,𝑥   𝑦,𝑅   𝜑,𝑎,𝑏,𝑟
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝑅(𝑎,𝑏)   𝐹(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑂(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑟,𝑎,𝑏)

Proof of Theorem rfovfvd
StepHypRef Expression
1 rfovfvd.f . . 3 𝐹 = (𝐴𝑂𝐵)
2 rfovd.rf . . . 4 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))
3 rfovd.a . . . 4 (𝜑𝐴𝑉)
4 rfovd.b . . . 4 (𝜑𝐵𝑊)
52, 3, 4rfovd 40225 . . 3 (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
61, 5syl5eq 2865 . 2 (𝜑𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
7 breq 5059 . . . . 5 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
87rabbidv 3478 . . . 4 (𝑟 = 𝑅 → {𝑦𝐵𝑥𝑟𝑦} = {𝑦𝐵𝑥𝑅𝑦})
98mpteq2dv 5153 . . 3 (𝑟 = 𝑅 → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}))
109adantl 482 . 2 ((𝜑𝑟 = 𝑅) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}))
11 rfovfvd.r . 2 (𝜑𝑅 ∈ 𝒫 (𝐴 × 𝐵))
123mptexd 6978 . 2 (𝜑 → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}) ∈ V)
136, 10, 11, 12fvmptd 6767 1 (𝜑 → (𝐹𝑅) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  𝒫 cpw 4535   class class class wbr 5057  cmpt 5137   × cxp 5546  cfv 6348  (class class class)co 7145  cmpo 7147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150
This theorem is referenced by:  rfovfvfvd  40227
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